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Doubting the estimate of Halfling Value

Discovering the value of a new kind of piece is one of the most difficult things in the field of chess variants. However, I find it an interesting problem and hope that I don't bore you too much by devoting so many words to it.

Funny Notation for Halflings

I want to use h, but it's already used, in a limited way. In fact, it can only appear in "fhN" or "bhN", that is, as a submodifier. thus it should be available for unambiguous use to mean "halfling" if a precaution is taken:

Because doubling the major letter is the notation for a rider, for example WW is the Rook, it makes sense to use "hh" for Halfling. Thus, the halfling R is hhR or hhWW.

"Worth Half a Piece"?

In various places I have said that adding a Ferz (or a Wazir or a Dabaaba or an Alfil) to a Knight produces a piece that is worth one and one-half Knights, that a Ferz as a piece by itself is worth half a Knight (although a Wazir or Dabaaba or Alfil as pieces by themselves are worth much less), and that a Halfling Bishop is worth half a Bishop.

I am puzzled by my apparent contradiction and wonder if I've made a mistake.

In Shantranj, the Knight was thought to be worth five Pawns (the difference being that Pawns could promote only to Ferz), and the Ferz was thought to be worth two. However, the difference between 40 percent of a Knight's value and half a Knight's value is just barely a third of a Pawn, one quantum of advantage, the same difference in value that some claim exists between Knight and Bishop. Thus it is fair to say that a Ferz is worth roughly half a Knight. Roughly is fair because experience has shown that the leveling effect and other factors make piece values fuzzy at best.

Adding a Ferz to a Knight adds half a Knight to the Knight. Once again, roughly will cover the situation: we would expect that because the Q is worth more than the separate R and B, the NF should be worth more than the separate N and F; but of course considering the proportions of the values involved, we would expect the NF to be worth less than N + F + half-a-Pawn; and because the Shatranj masters rated the Ferz as worth roughly 40% of a Knight, the sum of the values of N + F + half-a-Pawn is "roughly" a Knight and a half.

The Ferz as a piece by itself is worth roughly half a Knight, but on the low end of what can be described as half; the NF is worth roughly a Knight and a half, but on the high end of "half".

And what about the Halfling Bishop? It can always make any move a Ferz can make, and there is always at least one direction in which it can move further than the F. Mustn't it be wrong to say the F is worth half a Knight and also to say the Halfling B is worth half a B?

Although it's possible to invoke the same arguments as before and thus conclude that the hhB and the F are each worth "half a Knight" even though the hhB is clearly stronger, this question deserves a deeper look.

For a moment, let's consider the Knight and the Bishop. They are the same value, and yet the Bishop is worth a third of a Pawn more than a Knight; but in many positions the Knight is worth more than the Bishop. The contradictions in these statements do not shock us.

No two pieces with different moves can ever be exactly the same value; instead, the dynamics of Chess make all minor pieces have the same effective value, and likewise for the other ranks.

The F is a subset of the hhB, and so there are no positions where the F is worth more than the hhB; they are said to be in the same category, pieces worth half a Knight, one at the low end and one at the high end, and yet the difference in value between F and hhB is proportionately greater than the difference between B and N.

The estimate of the hhB as half a Bishop is probably pretty good, although it assumes that the hhB has the same value relative to the B as the hhR has to the R. The reason the estimate is probably good is that it is based on knowledge of the values of short Rooks, knowledge gained from and verified by experience with those pieces; and because the hhR is so similar in its characteristics to short Rooks that it seems reasonable to suppose that the average mobility calculation can be used to say that because the average mobility of the hhR is midway between that of the R2 and the R3, the value of hhR is also midway between R2 and R3.

The estimate of the hhB as half a Bishop is probably pretty good, although the hhB has not been extensively playtested. According to the theory that a Bishop is worth one third of a P more than a Knight, half a B is 1.67 Pawns; and according the the estimates of the Shatranj masters, a ferz ought to be worth 1.2 Pawns; and either is "roughly half a Knight", even though the difference is comparatively huge.

The average mobility of the F is 3.0625 and the average mobility of the hhB is 4.43, and the F is 69 percent of the hhB; the estimated value as above is 1.2 for F and 1.67 for hhB, so the F is 72 per cent of the hhB. Experience with the added value given by longer moves tells me that the ratio of value should be larger than the ratio of calculated mobility; and so I'd guess that the phrase "half a Bishop" is correct but imprecise.

In fact, correct but imprecise is what we want for values of chess pieces, because the values can never be really precise. If I say "half", you'll use your common sense; but if I say "fifty-five per cent", you may be blinded by the precision of the numbers; and even though 0.55 is probably correct, and a bit more precise than "half", for all practical purposes the difference in value between a piece worth 1.67 pawns and one worth 1.83 Pawns is negligible.

To sum up the conclusions, a Ferz as a piece by itself is worth less than half a Knight, but it's close enough that you can get away with calling it "half"; a Halfling Bishop is likely to be worth a bit more than half a B, maybe fifty-five per cent, but it's close enough to call it "half a Knight". The hhB may be worth half again as much as a Ferz, and so a piece such as NhhB would be too strong to be called an "Augmented Knight".